
Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} summarizes the effect of portfolio size on operating characteristics for five insurer portfolio sizes. Still, many subtleties cannot be adequately described. Tables~\ref{tab:InsurerPremiumsStandardErrorsLossRatioProbabilitiesbyPortfolioPortfolioSize0.01Increment} - \ref{tab:RiskLoadedReinsurancePremiumsByPortfolioSize} provide additional details, as insurer portfolio sizes change at a slower place. Thirty-five insurers, ranging in size from a single policyholder (e.g. An uninsured policyholder) to 309,000,000 policyholders as would be the cae with a national health insurer, were analyzed to show some of the subtleties in how insurer profits, losses, insolvency risks, surplus requirements, benefit levels, and reinsurance costs vary with portfolio size.

\subsection{Expanded Tables - Loss Ratio Probabilities} \label{sec:ExpandedTables-LossRatioProbabilities}

Tables~\ref{tab:InsurerPremiumsStandardErrorsLossRatioProbabilitiesbyPortfolioPortfolioSize0.01Increment} - \ref{tab:InsurerPremiumsStandardErrorsLossRatioProbabilitiesbyPortfolioPortfolioSize0.50Increment} show the probabilities of loss ratios higher than specified levels by insurer portfolio size. Within each table the evaluation points change from column to column, varying in amounts from one percent of insurer premiums to fifty percent of insurer premiums per column.

This makes it easier to see that very small insurers have a great deal of probability far above the Population Loss Ratio Estimate Cumulative Distribution Functions, while very large insurers have very little of the probability under their Population Loss Ratio Estimate Cumulative Distribution Functions. 
Looking at Table~\ref{tab:InsurerPremiumsStandardErrorsLossRatioProbabilitiesbyPortfolioPortfolioSize0.01Increment} we can see immediately that NHI, with 309,000,000 policyholders, has just 2 one hundredths of one percent of its probability higher than 0.7600 while an insurer with 10,000 policyholders has almost half of its probability above 0.7600. NHI is unlikely to ever incur claims costs higher than 0.7600 while all smaller insurers are very likely to incur high claims costs. Even an insurer with 10,000,000 policyholders has more than one-quarter of its probability above a loss ratio of 0.7600, showing that there are enormous advantages to letting a single, very large insurer manage risk compared to managing risk using large numbers of small, or very small, insurers.

\subsection{Expanded Tables - Profit Probabilities} \label{sec:ExpandedTables-ProfitProbabilities}

Tables~\ref{tab:InsurerPremiumsStandardErrorsProfitLossProbabilitiesbyPortfolioPortfolioSize0.01Increment} - \ref{tab:InsurerPremiumsStandardErrorsProfitLossProbabilitiesbyPortfolioPortfolioSize0.50Increment} show how portfolio size affects insurer's profitability. Tables~\ref{tab:InsurerPremiumsStandardErrorsProfitLossProbabilitiesbyPortfolioPortfolioSize0.01Increment} shows that NHI, the largest insurer, has probability 0.9998 that it will earn a profit of at least 9\% while an insurer with 1,000 policies has probability 0.5025 of earning profits of at least 9\%. 

Table~\ref{tab:InsurerPremiumsStandardErrorsProfitLossProbabilitiesbyPortfolioPortfolioSize0.05Increment} reproduces many of the results in Table~\ref{tab:InsurerOperatingResultsByPortfolioSize}, incrementing the changes in loss ratios by 0.05 per column. While all insurers have identical probabilities (0.5000) of profits greater than 10\%, Insurer E with 10,000 policyholders, has significant probabilities associated with losses of 5\%, 10\% and 15\% of 0.6179, 0.6554, and  0.6915 while insurers larger than PI have virtually no chance of incurring operating losses. If we accept the idea that there is negligible probability above four standard errors above the population loss ratio an insurer with 4,000,000 policyholders will not experience operating losses while all smaller insurers have some probability of incurring operating losses.


\subsection{Expanded Tables - Maximum Sustainable Benefits} \label{sec:ExpandedTables-MaximumSustainableBenefits}

Tables~\ref{tab:MaximumSustainableBenefitsPercentOfPremiums} and \ref{tab:MaximumSustainableBenefitInDollars} show the proportions of premiums that insurers can allocate to policyholder benefits and the dollar amounts of benefits per policyholder. Insurers can think ahead and adjust their benefits to achieve specific profitability, loss avoidance, and insolvency avoidance criteria. If all insurers want PI's probability (0.5000) of earning profits of at least 10\%, they can offer the same level of benefits to their policyholders as PI.

However, when all insurers seek PI's probability of profits of at least 5\%, or 0\%, smaller insurers must reduce benefits, while larger insurers can increase benefits for their policyholders. Larger, more efficient, insurers can offer significantly higher benefits to their policyholders, suggesting that if benefit provision efficiency is an important criteria by which to measure the success of our health care finance system, small numbers of large insurers are better benefit providers than small insurers.

The disparity in benefits grows as the comparison criteria moves further above the population loss ratio because smaller insurers have so much higher probabilities of incurring large losses than large insurers. If insurers want to match PI's probability of avoiding operating losses (0.9772), NHI can provide benefits of \$3,377 while an insurer with 20,000 policyholders can only provide benefits of \$572 per policyholder, per year.

When all insurers want to match PI's probability of avoiding operating losses of at least 5\% of premium revenues (0.9987), NHI can provide benefits of \$3,566 while an insurer with 30,000 policyholders can only provide benefits of \$136 per policyholder, per year and an insurer with 20,000 policyholders cannot consistently provide any level of benefits throughout the year without risking losing more than 5\% of its premium revenues.

\subsection{Expanded Tables - Surplus And Aggregate Surplus Requirements} \label{sec:ExpandedTables-SurplusAndAggregateSurplusRequirements}

Failed insurers do not meet their obligations to policyholders, claimants, employees, stockholders, or society. Preventing insurer failures is a critically important aspect of insurance regulation. If we assume that all insurers should be equally protected from insolvency (i.e. protected from insolvency with the same probability, regardless of size) Table~\ref{tab:SurplusRequirements1M} displays the surplus required to protect against Population Loss Ratio Estimates three standard errors above the Population Loss Ratio by portfolio size. 

However, the dollar value of surplus for single insurers is misleading since very large insurers do not require any insurance while very small insurers require very large amounts of surplus. The cutoff point, below which an insurer cannot expect to cover losses from current premium revenues is 2,250,000 policyholders. larger insurers need no surplus to meet the Solvency Preserving Loss Ratio solvency standard, but all small insurers need increasingly large amounts of surplus to meet this standard.

However, individual insurer surplus requirements do not tell the real story. Surplus funds are idle funds. They cannot be used for any other purpose except protecting the insurer's solvency. To meet this goal, surplus funds must be cash or highly liquid financial instruments that can be quickly turned into cash at, or about the face value. Government bonds are highly liquid but offer very low interest rates. This correspondence between low returns and low risk is characteristic of the type of investments insurers use to hold surplus funds. 

To see the magnitude of much money small insurers must idle, compared to large insurers, we need to compare the surplus requirements to insurer the entire population of 309,000,000 Americans. NHI can cover all 309,000,000, but we would need 309 PI's and 309,000 insurers, covering 1,000 policyholders each, to insure all 309,000,000 Americans. NHI can cover all Americans and it will not idle any money at all or 309,000 insurers can insure 1,000 policyholders each and would have to set aside almost 6 trillion dollars. If each American stood on their own and met the solvency standard, each American would have to set aside \$599,600 and an aggregate of over 185 Trillion in surplus.

Insurance is not just a good approach to risk management, it is necessary to sustain material well being in any industrialized economy. In the absence of insurance, each American would have to set aside so much money that our economy would grind to a halt. Most Americans would not be able to fully fund their personal health accounts over the course of their entire lives, and even those who could would have to forego a lot of personal luxuries in the process.

Even high deductible health insurance schemes, such as have been repeatedly enacted into law, suffer the same flaw. If each American had to set aside \$5,000 for a personal health account, the aggregate cost in idled funds would be more than 1.5 Trillion dollars. Diverting this level of assets from our economy for a benefit that can be provided with no diverted assets, is far too inefficient to be sound public policy.


\subsection{Expanded Tables - Risk Loaded Reinsurance Premiums} \label{sec:ExpandedTables-RiskLoadedReinsurancePremiums}

The core problem with capitation based health care finance mechanisms is that they lead to inefficient and under-compensated health care provider insurance operations. If health care providers were adequately compensated capitation financed health care would be unsustainable because the costs of health care providers' inefficient insurance operations would have to be paid by those transferring insurance risks to them.

Table~\ref{tab:RiskLoadedReinsurancePremiumsByPortfolioSize} shows the appropriate levels of risk loaded premiums an insurer should pay some other entity: insurer, reinsurer, or health care provider, to assume full responsibility for insurance policies it has written. If our Paradigm Insurer wants to transfer all its 1,000,000 policyholders to insurers that will have portfolio sizes, after the transfer, as listed in Table~\ref{tab:RiskLoadedReinsurancePremiumsByPortfolioSize}, PI should transfer different portions of each premium dollar based on the risk assuming entity's profit or loss avoidance goals and portfolio sizes.

When a very large insurer, such as NHI, assumes insurance risks from PI, its higher risk management efficiency allows it to accept less than 85\% of PI's premium revenues. As a result, both NHI and PI can virtually lock in profits as a result of the transfer. However, when PI transfers its insurance risks to smaller insurers, their inefficient risk management characteristics require higher payments from PI. If NHI is going to earn 5\% of PI's premiums, as profits, with probability 0.8413, PI need only pay NHI 80.28\% of its premium revenues, locking in profits of 4.72\% for itself.

However, if PI transfers sub-portfolios of insurance risks to insurers with 50,000 policyholders, after the transfer, PI should pay these insurers 102.36\% of its premium revenues for the transferred policyholders. Doing this would lock in losses of 17.36\% of PI's premium revenues while if it retains these risks PI has a probability of 0.5000 of earning profits greater than 10\% and a probability of earning profits of at least 5\% of 0.8413.

If PI transfers its insurance risks to very small insurers, 1,000 insurers each with 1,000 policyholders, PI should pay each insurer 238.11\% of its premium revenues and lock in a loss of 153.11\% on its insurance operations.

As was true with surplus requirements, policyholder benefits, and exposure to losses and insolvency, the smaller the insurer the more adverse the outcomes.